Determinat Bundles and Geometric Quantization Of Vortex Moduli Spaces ON Compact Kahler Surfaces

TL;DR

This paper constructs holomorphic determinant bundles on projective Kahler manifolds and demonstrates their role in geometric quantization of vortex moduli spaces, linking vortex and Seiberg-Witten moduli.

Contribution

It introduces a new approach to geometric quantization of vortex moduli spaces using determinant bundles on Kahler surfaces, extending previous work and connecting to Seiberg-Witten theory.

Findings

Existence of holomorphic determinant bundles serving as quantum line bundles

Moduli space of vortex equations is Kahler and projective under certain conditions

Quantization of Seiberg-Witten moduli via determinant bundles

Abstract

In this paper we first show that on projective manifolds (M, {\omega}), there are holomorphic determinant bundles (in the sense of Knusden-Mumford used by Bismut, Gillet, Soule) which play the role of the geometric quantum bundle, namely one for each input data of a Hermitian holomorphic line bundle L of non-trivial Chern class on a compact Kahler manifold Z (with Todd genus non-zero) and a choice of a geometric quantization of (M, {\omega}). Next we further study the generalization of the vortex equations on Kahler 4-manifold which has been studied earlier by Bradlow. We show that when the Kahler 4-manifold avoids some obstructions then the regular part of the moduli space is a Kahler manifold and admit a pull back of a Quillen determinant bundle as the quantum line bundle, i.e. the curvature is proportional to the Kahler form. Thus they can be quantized geometrically. In fact we show that the moduli space of the usual vortex equations on a projective Kahler 4-manifold is projective when the moduli space is smooth. Since in Kahler 4-manifold the vortex moduli and the Seiberg Witten moduli coincide our effort gives a quantization of Seiberg Witten moduli by determinant bundles